Cooperative binding systems are being studied taking into account site or submit interactions, ligand interactions, aggregation and redistribution in proteins and model systems. Methods are being developed to evaluate reasonable parameters describing these reactions. We study amino acids and proteins where the number of sites is small, and also those systems such as the protein-DNA complex and the hellix- coil transition where the number of sites can be very large. The usual method of analysis for large systems has involved approximations assuming an infinite number of sites. We, on the other hand, have developed exact combinatorial approaches to this problem. Our formalism analyzes both small and large numbers of sites and requires only reasonable times on the present high-speed computers. Generation of the exact polynomials describing these systems has permitted us to examine in detail properties of the reactions which were previously unknown. Our emphasis at present is on the analysis of the effect of linked functions (multiple ligands) and temperature on the cooperative process. - binding cooperativity ligand interactions subunit interactions aggregation